Quotient

In arithmetic, a quotient (from Latin: quotiens "how many times", pronounced /ˈkwʃənt/) is the quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as a fraction or a ratio. For example, when dividing twenty (the dividend) by three (the divisor), the quotient is six and two thirds. In this sense, a quotient is the ratio of a dividend to its divisor.

Notation

The quotient is most frequently encountered as two numbers, or two variables, divided by a horizontal line. The words "dividend" and "divisor" refer to each individual part, while the word "quotient" refers to the whole.

{\dfrac {1}{2}}\quad {\begin{aligned}&\leftarrow {\text{dividend or numerator}}\\&\leftarrow {\text{divisor or denominator}}\end{aligned}}{\Biggr \}}\leftarrow {\text{quotient}}

Integer part definition

The quotient is also less commonly defined as the greatest whole number of times a divisor may be subtracted from a dividend without the remainder becoming negative. For example, the divisor 3 may be subtracted up to 6 times from the dividend 20 before the remainder becomes negative:

20 − 3 − 3 − 3 − 3 − 3 − 3 ≥ 0,

while

20 − 3 − 3 − 3 − 3 − 3 − 3 − 3 < 0.

In this sense, a quotient is the integer part of the ratio of two numbers.

Quotient of two integers

The definition of a rational number is the quotient of two integers (as long as the denominator is not a zero).

More formal definitions:

A real number r is rational if, and only if, it can be expressed as a quotient of two integers with a nonzero denominator. A real number that is not rational is irrational.

Even more formally:

if r is a real number, then r is rational ⇔ ∃ integers a and b such that $r={\frac {a}{b}}$ and $b\neq 0$ .

The existence of irrational numbers – numbers that are not a quotient of two integers – was first discovered in geometry in such things as the ratio of the diagonal of a square to the side.

More general quotients

Outside of arithmetic, many branches of mathematics have borrowed the word "quotient" to describe structures built by breaking larger structures into pieces. Given a set with an equivalence relation defined on it, a "quotient set" may be created which contains those equivalence classes as elements. A quotient group may be formed by breaking a group into a number of similar cosets, while a quotient space may be formed in a similar process by breaking a vector space into a number of similar linear subspaces.